Gold nanoparticles as contrast agents for nonlinear microscopy

These nanoparticles can be potential contrast agents for nonlinear optical microscopy.. List of Abbreviations 3D Three dimensional 2PF Two photon fluorescence 3PF Three photon fluorescen

FOR NONLINEAR MICROSCOPY

NAVEEN KUMAR BALLA

(B.Tech, Indian Institute of Technology Madras)

A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

IN COMPUTATION AND SYSTEMS BIOLOGY (CSB)

SINGAPORE-MIT ALLIANCE NATIONAL UNIVERSITY OF SINGAPORE

2012

Acknowledgements

I started my doctoral studies with almost no knowledge of optics All I knew about light at that time was its speed, 299792458 m/s in vacuum I memorized this number from my physics course at high school and later used it as a password But I have enjoyed the five and half years of my grad life I was fortunate to meet some wonderful people during this time who were kind and helpful This is right opportunity to express my gratitude towards them

Firstly, I would like to thank my supervisors Prof Sheppard and Prof

So for their guidance and patience They gave me enough time to explore and learn things on my own Prof Sheppard spent most of his time with us in lab, explaining to us various optical phenomena and how they link with one another His passion for research was a great source of inspiration for us Prof

So is a great experimentalist Though I spent only a short time with him in lab, I greatly benefited from his knowledge about optical instrumentation His guidance in design and execution of experiments was valuable

During my doctoral studies, I learnt a lot of experimental skills from people at lab Elijah, Fa Ke, Dimitrios and Shakil frequently helped me with the optical set-up Wai Teng and Shalin helped me with Matlab coding Sounderya helped me with cell culture work and antibody conjugation Anupama taught me gold nanorod synthesis I would like thank these people and all others at the two labs

I would also like thank my big family back in India who has been very supportive all these years I would like to specially thank my parents and my sister for believing in me

I am grateful to Prof Jerome Mertz for his valuable comments on my manuscript

Finally I would like to thank SMA for the financial support and SMA office staff for helping us out with the paper work from time to time

Table of Contents

Declaration……… I

Chapter 1: Introduction

1.1 Motivation……… 1

1.2 Nonlinear Optics……… 2

1.3 Nonlinear Microscopy……… 8

1.4 SHG Microscopy……… 11

1.5 Gold Nanoparticles……… 12

1.6 Overview of Thesis……… 16

Chapter 2: Discrete Dipole Approximation for Second harmonic Scattering 2.1 Introduction……… 18

2.2 Theory……… 23

2.3 Results and Discussion……… 28

2.4 Conclusion……… 38

Chapter 3: Comparison between Coupled and Uncoupled Dipole Models for Nonlinear Scattering 3.1 Introduction……… 40

3.2 Theory……… 43

3.3 Results and Discussion……… 45

3.4 Conclusion……… 56

Chapter 4: Bio-inspired nano contrast agents for second harmonic generation microscopy 4.1 Introduction……… 58

4.2 Theory……… 62

4.3 Results and Discussion……… 66

4.4 Conclusion……… 77

Chapter 5: Surface Modification and Multiphoton Luminescence Microscopy

of Gold Nanorods

5.1 Introduction……… 79

5.2 Materials and Methods……… 84

Synthesis of gold nanorods……… 84

Pegylation of gold nanorods……… 86

Optimizing concentration of PEG……… 86

Protein / Antibody conjuation……… 87

Cell culture……… 87

Multiphoton Luminescence Imaging……… 88

5.3 Results and Discussion……… 89

5.4 Conclusion……… 103

Chapter 6: Conclusions……… 105

Chapter 7: Future Directions……… 109

Bibliography……… 112

Author’s Publications……… 131

Summary

Gold nanoparticles interact strongly with visible and near infrared wavelengths because of their shape dependent plasmon resonance These nanoparticles can be potential contrast agents for nonlinear optical microscopy But nonlinear scattering from small particles with different shapes is difficult to predict by analytical methods We have developed a numerical method which assumes the scatterer to be made of dipoles In our model, the dipoles of a scatterer interact with each other and with external radiation Previous dipole models for nonlinear scattering failed to take into account interaction between the dipoles We show here that the dipole coupling is necessary for predicting the effects of shape and size of a nanoparticle on its nonlinear optical properties The coupling between dipoles increases with increase in the magnitude of refractive index of the scatterer Similarly dipole coupling becomes important in regions where there is a sharp change in refractive index like edges

Gold nanoparticles synthesized by wet chemistry are generally symmetric in shape and therefore they are not good candidates of second harmonic generation (SHG) The coupled dipole model was used to design and optimize a gold nano-helix for SHG For a given excitation wavelength, the geometry of the helix can be tuned to yield maximum SHG The gold nano-helix was found to be 65 times better than a comparable gold nanorod

for SHG The approach for designing SHG scatterers can be extended to any other type nonlinear scattering

A generic methodology for modifying the surface of gold nanoparticles was developed Gold nanorods were used as sample gold nanoparticles Gold nanorods were coated with PEG to keep them stable in biological buffers The nanorods were conjugated with antibodies to target specific cell types The concentration of the antibody on the gold nanorods was optimized to reduce non-specific binding Multiphoton luminescence (MPL) microscope was used for imaging gold nanorods targeted to cancer cells When gold nanorods with longitudinal plasmon resonance (LPR) close to the laser wavelength (824 nm) were used, the nanorods got heated up very quickly even with 1 mW of excitation power But when long excitation wavelengths (1200 nm) were used, the heating of nanorods was significantly reduced and this allowed imaging for longer period of time Therefore longer excitation wavelengths, away from LPR of nanorods might be a better choice for MPL microscopy of gold nanorods

List of Figures

1.1 Jablonski diagram……… 5

1.2 Cartoon for SHG……… 5

1.3 TEM images of gold nanospheres and nanoshells……… 13

1.4 Confocal reflection image……… 14

1.5 SHG images of gold nanospheres……… 15

1.6 Nonlinear spectrum of gold nanorods……… 16

1.7 SHG image of gold nanosphere cluster……… 16

2.1 Cartoon of a sphere approximated as collection of dipoles…… 21

2.2 SHG scattering from a gold nanosphere……… 29

2.3 Schematic of experimental set-up for scattering………….……… 31

2.4 SHG from gold nanoparticles……… 33

2.5 SHG from silver nanoparticles……… 35

2.6 Experimental results of SHG from polystyrene beads………… 37

2.7 Simulation results of SHG from a polystyrene bead ……….…… 38

3.1 Focal field distribution……… 47

3.2 SHG induced at the focal point in collagen sheet……… 50

3.3 SHG from silver nanospheres (CDM Vs UDM)……… 51

3.4 THG from a polystyrene bead……… 52

3.5 CARS from a polystyrene bead……… 55

4.1 Cartoon of gold nano-helix……… 62

4.2 SHG as a function of pitch length……… 68

4.3 SHG as a function of elements of β……… 69

4.4 Extinction spectra of gold nano-helices……… 72

4.5 Extinction spectrum of a gold nanorod……… 74

4.6 SHG by a gold nano-helix and nanorod……… 77

5.1 Schematic of custom built multi-photon microscope……… 89

5.2 Absorption spectra of gold nanorods……… 90

5.3 Pegylation of gold nanorods……… 92

5.4 Absorption spectra of gold nanorods conjugated to antibody… 94

5.5 Z-stack of a cell……… 96 5.6 Targeting efficiency of gold nanorods……… 98 5.7 Photothermal damage to cells with 824 nm excitation………… 102 5.8 Photothermal damage to cells with 1200 nm excitation………… 103

List of Abbreviations

3D Three dimensional

2PF Two photon fluorescence

3PF Three photon fluorescence

CARS Coherent anti-Stokes Raman scattering

SHG Second harmonic generation

THG Third harmonic generation

PSF Point spread function

NA Numerical aperture

OCT Optical coherence tomography

HRS Hyper Rayleigh scattering

DDA Discrete dipole approximation

CDM Coupled dipole model

PDM Polarizable dipole model

UDM Uncoupled dipole model

NIR Near infrared

TMV Tobacco mosaic virus

CPMV Cowpea mosaic virus

LPR Longitudinal plasmon resonance

MPL Multiphoton luminescence

PEG Polyethylene glycol

CTAB Cetyl trimethylammonium bromide

DD Double distilled

SKBM Skeletal myoblasts

DMEM Dubelco’s modified eagle’s medium

PBS Phosphate buffer saline

EGFR Epidermal growth factor receptor

BSA Bovine serum albumin

α Molecular polarizability of first order

β Molecular polarizability of second order or first hyperpolarizability

γ Molecular polarizability of third order or second hyperpolarizability

Cext Extinction cross-section

Cabs Absorption cross-section

Jn Bessel function of nth order

Another important challenge for optical microscopy is source of contrast in biological samples Cells and tissues in general do not have

specific optical signatures which stand out of the background Moreover, the ability to monitor a specific component in the sample is important Therefore external contrast agents like fluorescent dyes or nanoparticles are necessary These contrast agents should be small enough to be able to target cells The optical signature from these contrast agents should be strong and it should allow for long term monitoring of the samples Hence there is a need for better contrast agent for nonlinear optical microscopy

1.2 Nonlinear Optics

Nonlinear optical microscopy refers to a collection of microscopy techniques which rely on nonlinear interaction between light and matter When light interacts with a material without changing the optical properties of the material, it is called a linear interaction Nonlinear interaction, on the other hand, occurs when the optical properties of the material are transiently changed by the light itself [2] To observe such nonlinear phenomena, a very high intensity of light is necessary and therefore almost all non-linear optics was developed after the invention of the ruby laser [3] However, antecedents

of nonlinear optics can be found in electro-optic effects like the Kerr effect [4] and the Pockels effect In the experiments leading to the discovery of these effects, a strong electric field was used to change the optical properties of a sample and a polarized light source was used to probe the extent of these

changes After the invention of laser, researchers were able to induce changes

in optical properties of some materials by using only light

There are a number of nonlinear optical phenomena but the ones important to microscopy are two photon fluorescence (2PF), three photon fluorescence (3PF), second harmonic generation (SHG), third harmonic generation (THG) and coherent anti-Stokes Raman scattering (CARS) 2PF was described theoretically by Maria Goppert-Mayer [5] in 1931 Before Goppert-Mayer’s work, it was believed that an electron could absorb only one photon to get excited to a higher energy level The light emitted on relaxation

of the excited electron to its ground state is called one (or single) photon fluorescence (1PF) (Fig 1.1(a)) Goppert- Mayer showed that an electron can absorb two photons to get excited to a higher energy level Two photon absorption is an intensity dependent absorption which can be described by equation (1.1)

0 2 .

a = a + a I (1.1)

where a is the total absorption coefficient, a0 is the linear absorption coefficient, a2 is the two photon absorption coefficient and I is the intensity of excitation light Generally, in order for the term a2I to be comparable in magnitude to a0, high intensities are required The fluorescence resulting from

two photon absorption is called 2PF (Fig 1.1(b)) The first experimental 2PF was observed from CaF2:Eu2+ crystals in 1961 [6] Three photon absorption is

an extension of this concept It should be noted here that the emitted photons are completely independent of how the excitation has occurred

The nonlinear optical phenomena like SHG, THG and CARS are types

of nonlinear scattering In SHG, two incident photons are jointly scattered by

an electron to give a single photon whose energy is the sum of energies of the incident photons (Fig 1.2) During this process, the refractive index of the sample is modulated by amplitude of the electric field (Eq 1.2)

Figure 1.1 Jablonski diagram of electronic transitions during (a)

one photon and (b) two photon fluorescence

Figure 1.2 Cartoon of second harmonic generation λ refers to

the wavelength of light e- represents an electron

Vibrational Levels

Ground state electronic level

Excited state electronic level

In any material, the elementary scatterer of light is an electron which is bound to a nucleus The polarization induced in a single electron-nucleus pair

E E p

E E p

Scattering is a coherent process which implies that the scattered light has well defined phase relation with the incident photons as well as with the sample The scattering of light, which is experimentally observed in the laboratory, is

a vectorial summation from different dipoles within a sample Therefore the structure of the sample and distribution of its optical properties play a very important role in the determining the nature of the scattered light The macroscopic scattering observed can be described by equation (1.5) which is expressed in terms of macroscopic properties:

1.3 Nonlinear Microscopy

2PF and SHG have proved to be excellent contrast mechanisms for imaging biological specimen The design of a nonlinear laser scanning microscope for SHG and 2PF was described by Sheppard and Kompfner [11] The first nonlinear laser scanning microscope was a second harmonic microscope, built

by Gannaway and Sheppard in 1978 [12] In this microscope, non-linear crystals were scanned through a focused laser beam and the SHG was recorded to form image pixel-by-pixel This set-up remains fundamentally unchanged even today, with the main difference being that a focused laser

beam is scanned across static samples in most cases The invention of femtosecond pulsed lasers with high repetition rates [13] gave a major boost

to the development nonlinear microscopy Femtosecond pulses provide high instantaneous intensities but keep the average power low The first 2PF microscope was built in Denk and coworkers [14] Starting in 1990s, nonlinear microscopy emerged as the most preferred imaging modality for thick biological samples Nonlinear laser scanning microscopy has intrinsic 3D imaging ability because the signal is generated only from the focal region Therefore, unlike a confocal microscope, there is no need for pinhole in nonlinear microscopes This makes the design of nonlinear microscopes much simpler The radial (r) and axial (z) point spread function (PSF) of a nonlinear microscope which uses nonlinear signal of order n can be calculated from equations (1.9) and (1.10) respectively [15]:

2 1

radial

J v PSF

u PSF

tissues Since spatial localization is a result of the nonlinear excitation process, the signal from the depths, though highly scattered, does not have to be descanned or passed through a pinhole Instead, the signal can be directly collected by a detector Both these factors improve the imaging depth of nonlinear microscopes In the case of 2PF microscope where fluorophores are involved, excitation of fluorophores occurs only in the focal region and hence photobleaching is reduced This allows for long term imaging with minimal loss in fluorescence signal.

The ability of 2PF and SHG microscopes to image deeper into the sample at sub-cellular resolution has contributed significantly to the development of various fields of biology [15-18] For example, Miller and coworkers [19] used 2PF to look into a lymph node of live mice and studied interactions of lymphocytes with the antigen presenting cells Sandoval and Molitoris [20] reviewed 2PF technique for studying the functioning of a mouse kidney Due to relatively high transparency of brain tissue to light, 2PF has been extensively used in neurobiology [16, 21] The technique is so benign

to biological tissue that in vivo nonlinear microscopy has been performed on human skin [22, 23] Unlike 2PF microscopy which mostly relies on external fluorophores for contrast, SHG microscopy mainly uses endogenous contrast

in biological tissue SHG microscopy has been used to image collagen, myosin and microtubule arrays in dividing cells [24] Collagen type I has been extensively imaged using SHG microscopy [25, 26] Detailed images of cornea

[27] and optic nerve head [28] were obtained because these structures are primarily made up of collagen Diagnosis of a fibrotic liver can be performed based on second harmonic images of the liver [29] Second harmonic imaging

of cell membrane stained with polar dye molecules has also been reported [30]

1.4 SHG Microscopy

SHG has certain advantages over 2PF Since no energy is absorbed in SHG, there are no issues of photo-bleaching or photo-toxicity Therefore SHG imaging can be performed over extended periods of time with no drop in signal intensity In 2PF, dye molecules can easily get saturated beyond a certain excitation power or pulse repetition rate This is one of the limitations

on signal intensity in 2PF microscopy SHG microscopy does not suffer from sample saturation as no real energy levels are involved SHG is a narrow band emission as compared to fluorescence The narrow band of SHG is also easy

to distinguish from the broad autofluorescence background in biological samples Quantum efficiency of fluorescent dyes which emit in the red region

of the spectrum is very low due to non-radiative decay This is a serious limitation for deep tissue imaging Red wavelengths are less scattered by tissue and therefore availability of efficient probes which emit in the red region of the spectrum is highly desirable There is no non-radiative decay in SHG and therefore SHG probes designed for red region of the spectrum

should be efficient The main limitation for SHG and THG in the infrared region is absorption by water which increases sharply for wavelengths above

1000 nm A new class of contrast agents for SHG microscopy has recently emerged [31] Some inorganic nanoparticles made of metals or metal oxides have been found to be strong scatterers for SHG Bariuam titanate (BaTiO3) nanocrystals have been used as probes for in vivo second harmonic imaging [32, 33] Strong SHG has been observed from zinc oxide (ZnO) nanocrystals Individual silver nanoparticle or in groups can be strong scatterers for SHG [34] Similarly SHG microscopy using gold particles has been reported [35]

1.5 Gold Nanoparticles

Gold nanoparticles have been extensively used as contrast agents for optical microscopy because of their superior optical properties, simple surface chemistry and biocompatibility Gold nanoparticles, like other metallic nanoparticles, have free electrons on their surface which oscillate collectively

at certain characteristic frequency surface plasmon resonance (SPR) frequency Gold nanoparticles absorb and scatter light strongly around their SPR frequency SPR of gold nanoparticles depend on the size For example, the SPR of gold nanospheres shows a red shift with increase in the size of the nanospheres [36, 37] Another way to change the SPR of gold nanoparticles is

by changing the shape of the particle [38] If we are looking at aggregates of gold nanoparticles, then again depending on the size and shape of the

aggregates, the optical properties change [39] The flexibility to tune the optical properties of gold nanoparticles make them good probes for optical microscopy Gold, being an inert metal, does not interact with the sample It has been shown in multiple studies than gold nanoparticles are not toxic to cells or tissues [40, 41] Gold nanospheres (Fig 1.3(a)) are the simplest form of gold nanoparticles Since these particles strongly scatter light they were used

as contrast agents for confocal reflectance microscopy (Fig 1.4) Antibody conjugated gold nanospheres were used to image cancer cells in culture [42]

as well as in ex vivo tissue [43] Gold nanoshells (Fig 1.3(b)) are better at scattering light in the near infrared region [44], and hence these nanoparticles have been used as contrast agents for optical coherence tomography (OCT) [45] Gold nanorods are another kind of gold nanoparticles which exhibit strong photoluminescence [46], and these nanoparticles have been used for multiphoton luminescence microscopy [47]

Figure 1.3 Transmission electron microscope images of gold

nanospheres (a) and gold nanoshells (b)

Figure 1.4 Confocal reflection image of dividing cells which

have been stained with gold nanospheres (Courtesy: Prof Colin

Sheppard)

Given the strong scattering ability of gold nanoparticles, it might appear that these nanoparticles would be promising contrast agents for SHG Unfortunately that is not the case Most common types of gold nanoparticles made in the laboratory have a symmetric structure that attenuates SHG Gold nanospheres dried on a coverslip can be imaged by SHG (Fig 1.5(a)) owing to the sharp change in refractive index at the point of contact with glass When this difference in refractive index is reduced by adding water, the intensity of SHG drops sharply (Fig 1.5(b))

Figure 1.5 Second harmonic images of 30 nm gold nanospheres

dried on a coverslip From top the particles were (a) open to air

and (b) submerged in DI water

Similarly, nonlinear spectra of gold nanorods (Fig 1.6) excited by femtosecond pulses centered at 800 nm show a small SHG signal at 400 nm and strong multiphoton luminescence, which includes three photon luminescence at wavelengths smaller than 400 nm and two photon luminescence at wavelengths greater than 400 nm This spectrum was acquired from a gold nanorod suspension in a cuvette The low magnitude of SHG can be attributed to the centrosymmetric shape of gold nanorods If the symmetry is broken, as in a cluster of gold nanospheres (Fig 1.7), strong SHG can be obtained These results agree well with hyper-Rayleigh scattering results from gold nanospheres [48] Hyper-Rayleigh scattering is an incoherent form of SHG

Figure 1.6 Nonlinear spectrum of gold nanorods when excited

with femtosecond pulses centered around 800 nm

Figure 1.7 Intense second harmonic signal from a cluster of gold

nanospheres The particles lie sandwiched between two layers

low melting agarose (0.5%)

1.6 Overview of Thesis

Conventional forms of gold nanoparticles are not good as contrast agents for SHG due to their symmetric structure Therefore asymmetric gold nanoparticles are required for SHG Such asymmetric gold nanoparticles are not readily available for experimental studies However it is possible to theoretically design asymmetric gold nanoparticles which will strongly scatter second harmonic light In other words, we can design artificial nonlinear

molecules [49] I have developed a numerical model to simulate nonlinear scattering from small structures of any arbitrary shape and size (Chapter 2)

My numerical model is derived from Discrete Dipole Approximation (DDA) [50] and another dipole model for nonlinear scattering [30] The previous dipole model for nonlinear scattering does not take into account interactions between dipoles I used the concept of dipole coupling from DDA to create a new dipole model for nonlinear scattering I have shown that my model can

be used for different types nonlinear scattering and that it gives better predictions than the previous model (Chapter 3) Using my model I have proposed the design of a chiral gold nanoparticle which will be a strong scatterer for SHG (Chapter 4) The design of this scatterer is based on a biological scaffold which makes the structure very robust Further, using gold nanorods I have developed methodology to functionalize gold nanoparticles

of any kind (Chapter 5) I have also demonstrated multiphoton luminescence

of cancer cells targeted with functionalized gold nanorods

of scattered light and the scattered light intensity is directly proportional to the incident light intensity However we are interested nonlinear scattering of light because of its applications to bioimaging as discussed in chapter 1 Nonlinear optical properties, specifically second harmonic generation (SHG) from small particles are highly dependent on the particle geometry It is well

known that under the dipolar approximation, SHG is forbidden from a centrosymmetric medium At an interface where this symmetry breaks, SHG can be observed However, in the case of spherical particles which are small relative to the excitation wavelength, SHG generated from one part of the particle surface is cancelled out by SHG generated from the other parts of the particle surface An expression for second order polarization in a centrosymmetric medium, proposed by Adler [51], forms the basis of most bulk models of SHG from small spheres Agarwal and Jha [52] proposed the first model for SHG from small metal spheres, predicting a dipolar bulk response and a quadrupolar surface response from the particles Others have taken into account the dynamics of electrons within a small sphere [53, 54] to predict its second harmonic scattering properties SHG from an array of quantum dots was calculated by modeling each quantum dot as a particle in a box which is being perturbed by excitation light [55] All the above works consider the bulk of the particle as a source of SHG The surface of a spherical particle can also cause SHG provided the particle is not small as compared to the excitation wavelength or if the excitation field is not uniform across the particle Under such conditions, the phase of the incident wave changes significantly as it travels across the particle As a result the surface SHG does not exactly cancel out The theory of SHG from small spheres under inhomogeneous illumination has been reported [56, 57] For spherical particles whose size is comparable to the excitation wavelength, the charge

induced on the surface by incident light can give rise to higher harmonics [58] Polar molecules immobilized on the surface of small polystyrene beads, when excited scatter second harmonic light which is polarized and has a specific angular distribution [59-61] Second harmonic response from small spheres with arbitrary surface response was calculated by Dadap et al [62, 63] All these analytical models give us an insight into the interaction of light with small particles While these models are important, they are restricted to spherical particles in a relatively homogeneous environment With the increasing importance of non-spherical metal nanoparticles being recognized [64-66], it is necessary to have a computational frame work that can efficiently predict the SHG properties of these non-spherical particles In addition to particle geometry, the illumination conditions may also be far from simple with significant field gradient or complex polarization distribution To add to this complexity, the local environment of the nanoparticles may be heterogeneous, especially for biomedical applications Numerical methods are better equipped to solve these problems

Here we use a numerical method called the discrete dipole approximation (DDA) [67] to calculate second harmonic scattering properties

of small particles This method was formulated to simulate linear scattering and absorption by interstellar dust particles [67, 68] In DDA, a scatterer is assumed to be made up of small polarizable dipoles (Fig 2.1) which interact among themselves and with the external field The optical properties of the

scatterer are given by the summation of the optical properties of all the constituent dipoles Draine and coworkers developed an efficient Fourier transforms based algorithm for DDA to facilitate fast computation of scattered fields by particles of various geometries [50, 69] The algorithm is implemented in FORTRAN language and it is available as a free program - DDSCAT[70] Hoekstra et al developed a parallel computing version of DDA which made it possible to simulate light scattering by large particles [71, 72] DDA has been used in different areas of research but with different names, like the coupled dipole method (CDM) [73] and the polarizable dipole model (PDM) [74]

Figure 2.1 Approximation of a sphere (a) as an arrangement of

sub-volumes on a cubic lattice (b) Each sub-volume behaves as

an individual dipole

One of the main advantages of DDA is its ability to handle arbitrary shapes of scatterers The method is easy to implement and it is computationally undemanding Although DDA was developed for materials

with moderate magnitudes of refractive index, it can be used to calculate scattering properties of metal nanoparticles which have high magnitudes of refractive index [75] The results predicted in the far field and in the highly sensitive near field regions agree well with results from other numerical methods As such, it has been extensively used to calculate linear optical properties of noble metal nanoparticles, especially for non-spherical shapes, clusters of nanoparticles and nanoparticles in heterogeneous local environments [39, 76-78] DDA has also been adapted to calculate scattering

by a wide variety of objects like periodic scatterers [79], photonic crystals [80], magnetic nanoparticles [81], metamaterials [82] and blood cells [83] Apart from scattering, this method has also been used to calculate optical forces on nanoparticles [84] All these applications of DDA consider linear interaction of light with matter We have shown here that by taking into account nonlinear interaction between light and matter, DDA can be extended to predict nonlinear optical scattering from small particles We have formulated the relevant mathematical expressions to calculate second harmonic scattering from small particles of different kinds We compared our computational results with experimental results reported earlier

2.2 Theory

When a nanoparticle is excited by an electromagnetic wave, a strong near field is created around the particle This strong field is ideal for nonlinear interaction of light and matter DDA has been used to calculate field enhancement around nanoparticles [76] but nonlinear scattering from small particles using this method has never been reported We have achieved this

by extending the DDA to calculate induced nonlinear dipoles within small particles when excited by light Here we briefly describe the linear model and then extend it to the nonlinear regime To begin with, a scatterer can be assumed to be made up of N small sub-volumes which are arranged on a cubic lattice If Einc,i is the incident field and P(1)i is the first order polarization induced at the center of the ith sub-volume, then these quantities are related by equation (2.1):

position vector of the ith dipole, then the expression for A(1)ij is given by equation (2.2):

ij i

ikr k r

, 1

dipoles fields One can also calculate extinction cross-section (Cext) and absorption cross-section (Cabs) of the scatterer [68]

(1)* (1) , 2

,

(1) ,

on the second order polarization, P(2) If βi is the first hyperpolarizability of the

number and not the excitation wave number Incidentally, similar models for SHG from surfaces were reported by Wijers et al [87] and Poliakov et al [88] Wijers et al assumed a silicon wafer surface to be made up of 20-80 stacked layers where each layer acts as a dipole In Wijers’ model, the geometry is simple – a linear arrangement of dipoles On the other hand, Poliakov et al [40] used their model to calculate various types of induced nonlinear polarization in molecules adsorbed on rough metallic surfaces The present

work, however, differs from these two earlier reports because here we have focused on far field second harmonic scattering properties of nanoparticles of different types and geometries Shape and size of a nanoparticle play important roles in its second order optical properties and we have demonstrated this effect by means of simple examples We have also shown here that our method can be used for nanoparticle composites Another dipole model [30, 89] worth mentioning here considers the dipoles to be independent

of each other and driven only by the incident field This model can be termed

as uncoupled dipole model and it is an approximation to our nonlinear coupled dipole model The coupled dipole model is a two step improvement

of the uncoupled dipole model The coupled dipole model first takes into account the corrected linear local field rather than the incident field itself Then it calculates the corrected second order polarization rather than obtaining it directly by squaring the linear local field

Since our model is an extension of DDA, it can be used for scatterers of random shapes The dipole size (d) should be such that the |m|kd < 1, where

m is the refractive index of the scatterer and k wave number of the scattering wavelength involved In the case of nonlinear scattering, multiple wavelengths are involved and therefore, we should consider the wavelength for which |m|k is maximum This ensures that the dipole size is good for all other wavelengths In our case, the second harmonic wavelength is the smallest wavelength The size of the dipole should also be small enough to

roughly approximate the shape of the scatterer This criterion is difficult to quantify, especially when the shape of the scatterer is irregular The refractive index of gold nanoparticles was obtained from Blanchard et al [90] The exact value of hyperpolarizability (β) of gold is not known However we can say that the surface of the particle is asymmetric and hence hyperpolarizability for dipoles on the surface is non-zero We have considered only one component

normal to the surface (Eloc,surf, ⊥) and yields a second order polarization in the same direction [91]:

2.3 Results & discussion

Based on the algorithm proposed by Draine and coworkers [69], we have developed a MATLAB code to implement the nonlinear DDA model Second harmonic scattering from a nanosphere is widely discussed in literature Therefore it is good sample scatterer to test our model We calculated the distribution of scattered second harmonic light from a gold nanosphere of 15

nm diameter The excitation source is a plane polarized wave of wavelength

800 nm, polarized along x-axis and propagating along positive z-axis The